Homogenization of a quasilinear parabolic equation with vanishing viscosity
The author considers the solution \(u^{\varepsilon}(t,x)\) of the following problem (\(\varepsilon > 0\)) \[ \begin{aligned} { \frac{\partial u}{\partial t} + \sum_{1=1}^N \, \frac{\partial A_i}{\partial x_i}} \Big({ \frac{x}{\varepsilon}}, u \Big) - \varepsilon \, \Delta u = f & \quad \text{in } {\mathbb R}^N \times [0,+\infty), \\ u(0,x) = u_0 { \Big( x,\frac{x}{\varepsilon}\Big)} & \quad\text{in } \mathbb{ R}^N \end{aligned} \] and study the asymptotic behaviour of the solutions as \(\varepsilon\) goes to zero. The functions \(A_i = A_i(y,v)\), \(i=1,\dots N\), are defined in \(Y \times {\mathbb R}\), \(Y\) the unit cell, are supposed to be \(Y\)-periodic and some growth conditions are required for the functions \(a_i(y,v) := \frac{\partial A_i}{\partial v}(y,v)\), \(1 \leqslant i \leqslant N\), and \(a_{N+1}(y,v) := - \sum_{i=1}^N \frac{\partial A_i}{\partial y_i}\). Under the assumption that \(u_0(x,y) = v(y, \bar{u}_0(x))\) for some \(\bar{u}_0\) the author proves that, as \(\varepsilon\) goes to zero, \[ u^{\varepsilon}(t,x) - v \big(x/\varepsilon, \bar{u}(t,x) \big) \to 0 \quad \text{in } L^2_{\text{loc}} \big( [0,+\infty) \times {\mathbb R}^N \big) \] where \(\bar{u}\) satisfies the scalar convervation law \[ {\frac{\partial \bar{u}}{\partial t} + \sum_{1=1}^N \, \frac{\partial \bar{A}_i}{\partial x_i}} \Big({\frac{x}{\varepsilon}}, \bar{u} \Big) = 0, \quad \bar{u}(0,x) = \bar{u}_0(x) \] for suitable \(\bar{A}_i\), defined by a cell problem. After a formal computation to obtain the homogenized problem using the usual asymptotic expansions, the proof is obtained using two-scale Young measures.
scientific article; zbMATH DE number 5129884
scientific article